The calculus of higher order subtyping, known as F ω ≤ , a higher-order polymorphic λ-calculus with subtyping, is expressive enough to serve as core calculus for typed object-oriented languages. The versions considered in the literature usually support only pointwise subtyping of type operators, where two types S U and T U are in subtype relation, if S and T are. In the widely cited, unpublished note [Car90], Cardelli presents F ω ≤ in a more general form going beyond pointwise subtyping of type applications in distinguishing between monotone and antimonotone operators. Thus, for instance, T U1 is a subtype of T U2, if U1 ≤ U2 and T is a monotone operator. My thesis extends F ω ≤ by polarized application, it explores its proof theory, establishing decidability of polarized F ω ≤. The inclusion of polarized application rules leads to an interdependence of the subtyping and the kinding system. This contrasts with pure F ω ≤ , where subtyping depends on kinding but not vice versa. To retain decidability of the system, the equal-bounds subtyping rule for all-types is rephrased in the polarized setting as a mutual-subtype requirement of the upper bounds.

We present a type system for a language based on F# , which allows certain type annotations to be elided in actual programs. Local type inference determines types by a combination of type propagation and local constraint solving, rather than by global constraint solving. We refine the previously existing local type inference system of Pierce and Turner[PT98] by allowing partial type information to be propagated. This is expressed by coloring types to indicate propagation directions. Propagating partial type information allows us to omit type annotations for the visitor pattern, the analogue of pattern matching in languages without sum types.

These notes belong to the course Type Systems for Programming Languages, given to fourth year students in Computing and Joint Mathematics and Computing with some experience in reasoning and logic, and students in the Advanced Masters programme at the Department of Computing, Imperial College, London. The course is intended for students interested in theoretical computer science, who possess some knowledge of logic. No prior knowledge on type systems or proof techniques is assumed, other than being familiar with the principle of induction. Aims • To lay out in detail the design of type assignment systems for programming languages. • To focus on the importance of a sound theoretical framework, in order to be able to reason about properties of a typed program. • To understand the concepts of: type checking, type reconstruction, polymorphism, type derivation, typeability, typing of recursive functions, termination in the context of typeability, and undecidable systems. • To study various systems and various languages, and to compare those and to select.